3.1287 \(\int \frac{x^4}{2 b+b x^5} \, dx\)

Optimal. Leaf size=13 \[ \frac{\log \left (x^5+2\right )}{5 b} \]

[Out]

Log[2 + x^5]/(5*b)

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Rubi [A]  time = 0.0030126, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {260} \[ \frac{\log \left (x^5+2\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(2*b + b*x^5),x]

[Out]

Log[2 + x^5]/(5*b)

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{x^4}{2 b+b x^5} \, dx &=\frac{\log \left (2+x^5\right )}{5 b}\\ \end{align*}

Mathematica [A]  time = 0.0029135, size = 17, normalized size = 1.31 \[ \frac{\log \left (b x^5+2 b\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(2*b + b*x^5),x]

[Out]

Log[2*b + b*x^5]/(5*b)

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Maple [A]  time = 0.001, size = 12, normalized size = 0.9 \begin{align*}{\frac{\ln \left ({x}^{5}+2 \right ) }{5\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^5+2*b),x)

[Out]

1/5*ln(x^5+2)/b

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Maxima [A]  time = 0.987855, size = 20, normalized size = 1.54 \begin{align*} \frac{\log \left (b x^{5} + 2 \, b\right )}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^5+2*b),x, algorithm="maxima")

[Out]

1/5*log(b*x^5 + 2*b)/b

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Fricas [A]  time = 1.70908, size = 27, normalized size = 2.08 \begin{align*} \frac{\log \left (x^{5} + 2\right )}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^5+2*b),x, algorithm="fricas")

[Out]

1/5*log(x^5 + 2)/b

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Sympy [A]  time = 0.191152, size = 8, normalized size = 0.62 \begin{align*} \frac{\log{\left (x^{5} + 2 \right )}}{5 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x**5+2*b),x)

[Out]

log(x**5 + 2)/(5*b)

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Giac [A]  time = 1.19999, size = 22, normalized size = 1.69 \begin{align*} \frac{\log \left ({\left | b x^{5} + 2 \, b \right |}\right )}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^5+2*b),x, algorithm="giac")

[Out]

1/5*log(abs(b*x^5 + 2*b))/b